I prove that $a$ and $b$ are parallel but can't prove that $c$ and $d$ are parallel.
The angles, which are $135^o$ and $45^o$ are alternate angles.They aren't equal , so $c$ isn't parallel to $d$ but the answer is that $c$ is parallel to $d$. Am I right that they aren't?

Use corresponding angles to see if the lines are parallel. Note if the alternate interior angles are congruent. This will prove if any of these two pairs might fall under the category of Euclidean parallel lines.